We study the symplectic mapping class groups of (CP2#5CP2, ω). Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures J0 as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by Wu [Math. Ann. 359 (2014), pp. 153–168]) and the study of a semi-toric model to understand a “connecting map”, whose cokernel is the symplectic mapping class group. Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms ω. Let Nω be the number of (−2)-symplectic spherical homology classes. Torelli SMCG is trivial if Nω > 8; it is π0(Diff+(S2, 5)) if Nω = 0 (by Seidel [Lecture notes in Math., Springer, Berlin, 2008] and Evans [J. Symplectic Geom. 9 (2011), pp. 45–82]); and it is π0(Diff+(S2, 4)) in the remaining case. Further, we completely determine the rank of π1(Symp(CP2#5CP2, ω)) for any given symplectic form. Our results can be uniformly presented in terms of Dynkin diagrams of type A and type D Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds (open problem 16 in McDuff-Salamon’s book 3rd version [Oxford graduate texts in mathematics, Oxford University Press, Oxford, 2017]).
|Original language||English (US)|
|Number of pages||54|
|Journal||Transactions of the American Mathematical Society|
|State||Published - 2022|
Bibliographical noteFunding Information:
Received by the editors February 10, 2020, and, in revised form, June 12, 2021, and July 8, 2021. 2020 Mathematics Subject Classification. Primary 57R17, 53D35; Secondary 14D22, 57S05. The first author was supported by NSF Grants and an AMS-Simons travel grant. The second author was supported by NSF Grants. The third author was partially supported by Simons Collaboration Grant 524427.
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